The embodiments described herein relate generally to correction methods X-ray diffraction profiles and, more particularly, to energy-based and angular-variation-based correction methods for X-ray diffraction profiles.
At least some known detection systems are used at travel checkpoints to inspect containers, such as carry-on luggage and/or checked luggage, for concealed contraband, such as weapons, narcotics, and/or explosives. At least some such detection systems include X-ray imaging systems. An X-ray imaging system includes an X-ray source that transmits X-rays through a container towards a detector. An output of the detector is processed to identify a set of objects and/or materials within the container. In addition, at least some known detection systems include X-ray diffraction imaging (XDi) systems. At least some known XDi systems use inverse fan-beam geometry (a large source and a small detector) and a multi-focus X-ray source (MFXS) to detect objects and/or materials. Further, some known XDi systems provide an improved discrimination of materials, as compared to that provided by other known X-ray imaging systems, by measuring d-spacings between lattice planes of micro-crystals in materials. X-ray diffraction (XRD) may also yield data from a molecular interference function that may be used to identify other materials, such as liquids, in the container.
Some known energy-dispersive XRD profiles that are measured with an electron impact X-ray source are normalized against an emission spectrum of the X-ray source to remove spectral structures, such as anode K characteristic lines and/or a non-uniform form of bremsstrahlung background. However, such known normalization significantly increases noise in the XRD profile, even when a primary spectrum of the X-ray source is substantially noise free. Further such known normalization suppresses genuine XRD peaks within a region where a bremsstrahlung curve has its peak intensity.
A non-uniform emission spectrum of an electron impact X-ray source, or primary spectrum, is directly reflected in a shape of its energy-dispersive XRD profile. Such non-uniformity of the emission spectrum arises in two separate ways. First, a K alpha line, a K beta line, and a K edge of an anode of the X-ray source are characteristic effects arising from an interaction of high-energy electrons from a cathode with K shell electrons orbiting nuclei of the anode. Second, the bremsstrahlung or continuous component of the emission spectrum has non-uniformity originating from an interaction of high-energy electrons from the cathode with an electric field of the anode nuclei. In order to avoid adverse effects of the non-uniform primary spectrum on measured XRD spectra, the measured XRD spectra are often normalized against the primary spectrum.
One known process of normalizing the measured XRD profile against the non-uniform primary spectrum essentially multiplies the measured XRD profile by varying weighting factors. For example, if a signal in a k-th channel of the measured spectrum is Sk and a weighting factor for this channel is wk, a noise-to-signal ratio (NSR) can be calculated using the following equation, where RMS denotes the root mean square deviation:
                                          RMS            ⁢                                                  ⁢            Noise                    Signal                =                                                            [                                  〈                                                            ∑                      k                                        ⁢                                                                                  ⁢                                                                  S                        k                                            ⁢                                                                                          ⁢                                              w                        k                        2                                                                              〉                                ]                                            1                ⁢                                  /                                ⁢                2                                                    〈                                                ∑                  k                                ⁢                                                                  ⁢                                                      S                    k                                    ⁢                                                                          ⁢                                      w                    k                                                              〉                                .                                    (                  Eq          .                                          ⁢          1                )            
For Poisson statistics that govern photon scattering experiments, a minimum NSR is achieved when all of the weighting factors are equal to one another. As such, a compromise is made when correcting for the disturbing effects of the primary spectrum on the measured XRD profile. On one hand, removing non-uniformities from the measured XRD profile improves feature extraction and/or substance identification. On the other hand, known normalization processing effectively increases noise in the measured XRD data, especially for lower density objects, even when the primary spectrum form is substantially noise free. As such, known normalization processing may reduce detection rates and/or increase false alarms.
Further, at least some known detection systems have a multiple-inverse fan-beam (MIFB) XDi topology, which has approximately 10% variation in an angle of scatter during a scan. Such scatter angle variations affect both the abscissa (momentum) and ordinate (intensity) scales of measured spectra. Further, at least some known XRD profile databases used with MIFB XDi systems include XRD profiles of known substances acquired at a reference scatter angle θR. When the actual scatter angle of the measured spectra vary from the reference scatter angle θR, the scanned object may not be accurately identified.
In known MIFB XDi systems, a certain object voxel is traversed by primary rays from several different X-ray source foci I directed to corresponding detector elements J. An actual angle of scatter θIJ is given by the relationship:θIJ=θ·cos γ  (Eq. 2)where γ is an angle that a primary ray from the I-th focus to the J-th detector element makes with an X-axis, and θ is an ideal scatter angle. In one example, cos γ has a minimum value of ˜0.9 and, as such, the actual angle of scatter θIJ is reduced relative to its maximum value by this proportion.
The relationship between momentum transfer x and the actual scatter angle θ is:
                    x        =                              E                          h              ⁢                                                          ⁢              c                                ·                      sin            ⁡                          (                              θ                2                            )                                                          (                  Eq          .                                          ⁢          3                )            where E is photon energy, h is Planck's constant, and c is the speed of light. As such, the correct value of θIJ from Equation 2 is incorporated into Equation 3 to correctly transform the photon energy scale E into a momentum scale x. However, because the primary spectrum is non-uniform, it is not sufficient to only transform the energy scale E into a correct momentum scale x. More specifically, the photon intensity at energy E1 corresponding to a momentum x for an angle θ1 is different than the photon intensity at energy E2 corresponding to the same momentum x for an angle θ2. However, the above-described known method does not transform a photon intensity scale to produce measured XRD profiles that are comparable to the reference XRD profiles.